where γ⁢(z0,z) is a path entirely contained in G
with initial point z0 and final point z.

The functionF:G→ℂ is well defined.
In fact, let γ1 and γ2
be any two paths entirely contained in G
with initial point z0 and final point z;
define a circuit Γ by joining γ1 and -γ2,
the path obtained from γ2 by
“reversing the parameter direction”.
Then by linearity and additivity of integral

∫Γf⁢(w)⁢𝑑w=∫γ1f⁢(w)⁢𝑑w+∫-γ2f⁢(w)⁢𝑑w=∫γ1f⁢(w)⁢𝑑w-∫γ2f⁢(w)⁢𝑑w;

(2)

but the left-hand side is 0 by hypothesis,
thus the two integrals on the right-hand side are equal.

We must now prove that F′=f in G.
Given z∈G, there exists r>0
such that the ball Br⁢(z) of radius r centered in z
is contained in G.
Suppose 0<|Δ⁢z|<r:
then we can choose as a path from z to z+Δ⁢z
the segmentγ:[0,1]→G parameterized by t↦z+t⁢Δ⁢z.
Write f=u+i⁢v with u,v:G→ℝ:
by additivity of integral and the mean value theorem,